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The probability distribution of the current in the asymmetric simpleexclusion process is expected to undergo a phase transition in the regime ofweak asymmetry of the jumping rates. This transition was first predicted byBodineau and Derrida using a linear stability analysis of the hydrodynamicallimit of the process and further arguments have been given by Mallick andProlhac. However it has been impossible so far to study what happens after thetransition. The present paper presents an analysis of the large deviationfunction of the current on both sides of the transition from a Bethe ansatzapproach of the weak asymmetry regime of the exclusion process.

Bethe Ansatz for the Weakly Asymmetric Simple Exclusion Process and phase transition in the current distribution